Local linear regression is used in a variety of data fitting applications. Particular applications within the realm of color imaging include printer and scanner characterization. A typical regression problem involves first gathering a training set of input data points from an input space and corresponding output data points from an output space. For the color characterization application, both input and output spaces are multi-dimensional color spaces. The goal of the regression algorithm is then to derive mappings from every point in the input space to the output space while minimizing error over the training set. An additional consideration is to ensure that the regression does not overfit the data in the sense that it is robust enough to filter out noise in the training data. Local regression algorithms are often used in situations where a single global fit may be inadequate to approximate complex non-linear transforms, as is typical in printer characterization. Instead, local transforms are derived where the regression parameters vary as a function of the input data point. Locality in regression is achieved by using a weighting in the error minimization function which varies (typically decays) as a function of the distance from the regression data point. Choice of these weight functions is typically intuitively inspired, and not optimized for the training set. This sometimes results in large regression errors especially with sparse training data. A key fundamental question hence remains on how to best use a certain local neighborhood of data points in regression problems and how to quickly compute optimally local transforms using certain local neighborhoods of data points in regression problems.
Incorporation By Reference
The following references are totally incorporated herein by reference.
R. Bala, “Device Characterization,” Digital Color Imaging Handbook, Chapter 5, CRC Press, 2003.
C. G. Atkeson, A. W. Moore and S. Schaal, “Locally weighted learning,” Artificial Intelligence Review, 1997.
E. Chong, S. Zak, “An Introduction to Optimization,” 2nd Ed, Wiley, 2001.